Fracture Mechanics of Concrete and Concrete Structures Recent Advances in Fracture Mechanics of Concrete - B. H. Oh, et al.(eds)
ⓒ 2010 Korea Concrete Institute, Seoul, ISBN 978-89-5708-180-8
Surface energy and fracture energy
F. H. Wittmann
Aedificat Institute Freiburg, Germany, and Qingdao Technological University, Qingdao, China
Z. Sun
Qingdao Technological University, Qingdao, China; now Binzhou Finance Bureau, Binzhou, China
T. Zhao
Qingdao Technological University, Qingdao, China
ABSTRACT: Properties of cement-based materials are governed by their nano-structure in general and by the
interaction between the surfaces of the nano-particles with their hygral surrounding in particular. In this contribution local damage and nano-crack formation in the three-dimensional network of nano-particles is considered to be at the origin of crack formation. Macroscopic fracture energy can be considered to be the sum of
a huge number of small energies consumed by nano-crack formation. Considering the physical process of
nano-crack formation it is possible to predict the influence of variable environments on fracture energy. This
hypothesis is verified by experimental investigations. The influence of aqueous NaCl solutions on strength,
fracture energy and swelling has been determined. It is shown that values of fracture energy as determined in
the laboratory are not conservative if applied for prediction of crack formation of reinforced concrete structures situated in marine environment. This fact is of major importance for durability and service life of reinforced concrete structures in marine environment.
1 INTRODUCTION
Fracture energy of different types of concrete has
been measured by many authors by different techniques, and for many years by now (see for example
Carpinteri et al. 2007). In the literature there are various standard test methods described such as direct
tension test, three point and four point bending test,
and wedge splitting test, which all may serve to determine fracture energy experimentally. In the fictitious crack model by Hillerborg (Hillerborg et al.
1976, Hillerborg 1985) fracture energy of concrete is
introduced in a phenomenological way. The overall
energy consumed by crack formation in sufficiently
large specimens is considered to be a material property. Crack formation induced by shrinkage for instance can be predicted with this approach in a realistic way (Alvaredo 1994). In this global approach,
however, physical processes involved in crack formation are neglected. For this reason the influence
of changes of the environment on fracture energy
cannot be taken into consideration directly and quantitatively. If a global approach is used, it is not clear
and not of primary interest to know precisely what
happens in the fracture process zone when a crack is
propagating.
Conventional concrete is a composite material
with mineral aggregates embedded in a cement-
based matrix. Most characteristic properties such as
shrinkage or swelling and compressive or tensile
strength are governed by the nano-structure built up
by hydration products of cement. When a load is applied to the three-dimensional framework of nanoparticles, a complex state of stress with local stress
concentrations is built up. At sites where the stresses
are above the local load bearing capacity, damage
will occur by nano-crack formation. This process
progressive damage leading to macroscopic cracks
can be modeled numerically.
In this contribution the macroscopically observed
fracture energy shall be linked with the elementary
processes in the nano-structure. It will be shown that
in this way it is possible to describe the influence of
chemical composition of the pore solution on
strength, fracture energy, and shrinkage or swelling
at least phenomenologically.
2 FUNDAMENTALS
In a homogeneous ideal brittle material the Griffith
criterion can be applied to describe crack formation.
If we consider a nano-crack in the xerogel of the
hardened cement paste with the length of 2c the critical stress σ is given by the following equation:
= − D ( h , T ) ∇h
2 Eγ
σ =
(1)
(1)
c
The πproportionality
coefficient D(h,T) is called
moisture permeability and it is a nonlinear function
(1) E stands
the elasticTmodulus
of In
theEquation
relative humidity
h andfortemperature
(Bažant
of
the
material
around
the
crack
and
γ
is
the
surface
& Najjar 1972). The moisture mass balance requires
energy
and
at
the
same
time
the
fracture
energy
on
that the variation in time of the water mass per unit
the
nano-scale.
Formation
one nano-crack
be
volume
of concrete
(water of
content
w) be equalcan
to the
considered
to the
be moisture
the elementary
divergence of
flux J step of accumulation of damage and finally crack formation. To create∂ a macroscopic crack a huge number of nano(2)
− = have
∇ • J to be formed first.
cracks
∂
If an external load is applied, early nano-cracks
are The
formed
a macroscopic
as low
as sum
one
wateratcontent
w can be stress
expressed
as the
third
of
the
load
bearing
capacity.
With
increasing
of the evaporable water we (capillary water, water
load
and morewater)
cracksandaretheformed
and the
vapor,more
and adsorbed
non-evaporable
length
of
existing
cracks
increases
(see
for
example
(chemically bound) water wn (Mills 1966,
Zaitsev
& Wittmann
induces
Pantazopoulo
& Mills1981).
1995).This
It isprocess
reasonable
to
damage
to
the
material,
which
can
be
observed
assume that the evaporable water is a functionbyofa
gradual
stiffnessoforhydration,
by increased
and
relative decrease
humidity,of hthe
, degree
αc, capillary
absorption
(see
for
example
Wittmann
degree of silica fume reaction, αs, i.e. we=we(h,αetc,αal.s)
2006).
Once the maximum
stress is reached isotherm
the den= age-dependent
sorption/desorption
sity
of
nano-cracks
is
high
enough
and
micro-cracks
(Norling Mjonell 1997). Under this assumption and
are
by coalescence
by formed
substituting
Equation of
1 nano-cracks.
into EquationThe2 denone
sity
of
nano
and
micro-cracks
is
particularly large in
obtains
the so-called fracture process zone where the macroscopic
crack will be formed
finally.
∂w
∂w
∂w ∂h
e
e α&is+certainly
e
(1)
&
α
w&n
• ( D ∇h) = Equation
− On the+ ∇nano-scale
(3)a
c +
s
h
∂
α
∂
α
∂
∂
h
t
good approximation to describe
nano-crack
formac
s
tion or damage. Then the surface energy γ is the energy
needed
one link
in sorption/desorption
the nano-structure.
the
slope
of
the
where
∂we/∂htoisbreak
This
value (also
dependscalled
on the moisture
humidity ofcapacity).
the surroundisotherm
The
ing
atmosphere.
Under
usual
conditions
a water
film
governing
equation
(Equation
3)
must
be
completed
isbyadsorbed
onboundary
the surface
of
gel conditions.
particles with a
appropriate
and
initial
characteristic
thickness
Γ,
which
depends
on the RH
The
relation
between
the
amount
of
evaporable
of
the surrounding
air.
The reduction
of ‘‘adsorption
surface enwater
andto relative
humidity
isfilm
called
ergy
due
an
adsorbed
water
is
shown
in Figisotherm”
if measured
with
increasing
relativity
ure
1
(Setzer
1972).
The
surface
energy
γ
0 is at
humidity and
‘‘desorption
isotherm”
in Under
the opposite
maximum
in
the
completely
dry
state.
these
case.
Neglecting
their
difference
(Xi
et
al.
1994),
in
conditions
strength
is at isotherm”
maximum will
too be
as used
predicted
the
following,
‘‘sorption
with
by
the Munich
model
(Wittmann
1976, 1977).
reference
to both
sorption
and desorption
conditions.
By the way, if the hysteresis of the moisture
isotherm would be taken into account, two different
relation, evaporable water vs relative humidity, must
be used according to the sign of the variation of the
relativity humidity. The shape of the sorption
isotherm for HPC is influenced by many parameters,
especially those that influence extent and rate of the
chemical reactions and, in turn, determine pore
structure and pore size distribution (water-to-cement
ratio, cement chemical composition, SF content,
curing time and method, temperature, mix additives,
etc.). In the literature various formulations can be
found to describe the sorption isotherm of normal
concrete (Xi et al. 1994). However, in the present
paper the semi-empirical expression proposed by
Norling Mjornell (1997) is adopted because it
J
explicitly accounts for the evolution of hydration
reaction and SF content. This sorption isotherm
reads
we (h α c α s ) =
,
,
⎡
G1 (α c , α s )⎢⎢1 −
⎢
⎣
⎤
⎥
− α c )h ⎥⎥
⎦
∞
(g α
c
e
∞ − α )h
⎡
(g α
c
c −
⎢
K (α c α s ) e
⎢
10
10
,
1
w
1
1
⎣
t
Proceedings of FraMCoS-7, May 23-28, 2010
1
+
(4)
⎤
1⎥
⎥
⎦
where the first term (gel isotherm) represents the
physically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillary
water. This expression is valid only for low content
of SF. The coefficient G1 represents the amount of
water per unit volume held in the gel pores at 100%
relative humidity, and it can be expressed (Norling
Mjornell 1997) as
c α c+ ks α s
G (α c α s ) = k vg
c vg s
(5)
,
1
where kcvg and ksvg are material parameters. From the
maximum amount of water per unit volume that can
fill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtains
w
0
⎡
⎢
−
⎢1 −
1
⎢
⎣
α s + 0.22α s G
− 0.188
K (α c α s ) =
,
c
s
⎛
10⎜
⎝
1
e
⎛
10⎜
e
g αc − αc h
−
∞
1
⎞
⎟
⎠
⎝
g α c∞ − α c ⎞⎟h ⎤⎥
1
⎠
⎥
⎥
⎦
(6)
1
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
free (evaporable) water content in concrete at
various ages (Di Luzio & Cusatis 2009b).
2.2 Temperature evolution
Note that, at early age, since the chemical reactions
Figure
1. Decrease
surface energy
γ in Equation
as funcassociated
with ofcement
hydration
and SF(1)reaction
tion
relative humidity
p/p0 (Setzer 1972).
are ofexothermic,
the temperature
field is not uniform
for non-adiabatic systems even if the environmental
If we assume
that by changing
the moisture
temperature
is constant.
Heat conduction
canconbe
tent
all
values
with
the
exception
of
γ
remain
described in concrete, at least for temperatureconnot
stant
we obtain
the following
equation
the resultexceeding
100°C
(Bažant &
Kaplanfor 1996),
by
ing
change
of
strength:
Fourier’s law, which reads
= −2λ∇Tγ h
(7)
(2)
where
is the heat flux, T is the absolute
temperature,
andforλ the
is the
heat conductivity;
in this
where σh stands
strength
at relative humidity
h and σ0 for the strength in vacuo, γh stands for the
⎛qσ h ⎞
⎜
⎟
⎝ σo ⎠
=
γo
q
= 1−
∆γ
γo
surface energy at relative humidity h and γ0 the surface
energy in vacuo, ∆γ is the change of surface energy.
The Munich model also predicts a length chance
if the moisture content changes (Wittmann 1976,
1977). The length change ∆l of a specimen with
length l0 can be described by means of the Bangham
equation (Bangham & Fakhoury and Bangham &
Razouk):
∆l
lo
= λ (γ o − γ h )
(3)
where λ is a material constant, which takes the stiffness of the material into consideration. By combining Equation (1) and Equation (3) we obtain:
(
σh 2
1 ∆l
) = 1−
σo
λγ o lo
(4)
That means that the square of the related strength
is linearly related to swelling. These relations are
applicable to nano-porous materials in the range of
RH < 60 %. In this range strength as well as swelling or shrinkage are controlled essentially by the
change of surface energy. At higher relative humidity disjoining pressure dominates and should be
taken into consideration separately (Wittmann 1976).
So far we have considered formation of one single nano-crack in hardened cement paste. When the
density of nano-cracks becomes high enough microcracks are formed and finally macro-cracks. The energy necessary to form a macro-crack is obviously
the sum of all nano-cracks which are necessary to be
formed before failure. The Griffith criterion is not
applicable in its simple from given by Equation (1)
to cement based composite materials. But still surface energy γ is at the origin of all the elementary
processes leading to crack formation. The sum of all
small quantities of surface energy needed to form
the huge number of nano-cracks necessary to create
a real crack finally is the macroscopic fracture energy,
which we determine with experimental methods.
Based on these considerations we can state that
adsoption of water in the nano-structure reduces surface energy of gel particles and in an indirect way
fracture energy as measured on the macro-scale. In
marine environment salt water can penetrate into the
pore space of concrete. As seawater and salt water in
general have a higher surface energy as compared to
water, we may expect that salt water and seawater
reduce strength even more than pure water. In general strength and hygral length chance depend on the
chemical composition of the pore solution.
3 EXPERIMENTAL
Preliminary results have been published recently
J = − D (In
h, Tthis
)∇h contribution new re(Wittmann et al. 2007).
sults shall be presented to broaden the experimental
basis, which serves The
as verification
of thecoefficient
above de- D(h,T)
proportionality
scribed relations. moisture permeability and it is a nonlinea
Tests have been
out on
concreteh and
and temperature
morof carried
the relative
humidity
tar specimens. For
concrete
broken
natural
gravel
& Najjar 1972). The moisture mass balanc
with a maximum that
diameter
of 20 mm
and local
the variation
in time
of theriver
water mas
sand with a maximum
diameter
of
5
mm
were
used.w) be eq
volume of concrete (water content
In order to obtaindivergence
good workability
1% of flux
a plastiof the moisture
J
cizer related to the mass of cement was added. Ordinary Portland cement
type 32.5 was used through− ∂ of= ∇concrete
•J
out. The composition
is given in Table 1.
∂
w
t
Table 1. Composition of concrete in kg/m3.
The
water content
w can be expressed a
Cement Gravel of the
Sandevaporable
Waterwater w
W/C
320
1267
653
160
0.5e (capillary wa
vapor, and adsorbed water) and the non-e
3
water wn (Mil
Table 2. Composition(chemically
of mortar in kg/mbound)
.
Pantazopoulo
&
Mills
1995). It is reas
Cement
Sand
Water
W/C
assume
that
the
evaporable
450
1350
225
0.5 water is a fu
relative humidity, h, degree of hydration
degree
silica fumehave
reaction,
s, i.e. we=w
Two types of concreteofspecimens
been αpre=
age-dependent
sorption/desorption
pared: (a) slabs with the dimensions of 400 x 400 x
(Norling
Mjonell
36 mm for wedge
splitting
tests 1997).
(WST)Under
and this
(b) assum
by substituting
1 into
beams with the dimensions
of 100Equation
x 100 x 510
mm Equati
obtains
for three point bending tests (3PB). For tests to determine compressive strength cubes with an edge
∂w
∂w been
length of 200 mm have
e α& + ∂we α& + w
e ∂h + ∇cast.
( D ∇h ) =
•
−
c
s
In addition mortar
a maximum
h
∂α aggregate
∂α
∂h with
∂t
c
s
size of 5 mm has been prepared using the same type
of cement. The composition of the mortar is indi/∂h is the
of thetwo
sorption/
∂wemortar
cated in Table 2. where
From the
theslope
following
isotherm
(also
called
moisture
types of specimens have been produced: (a) slabs capac
governing
equation
3) must be
with the dimensions
400 x 400
x 36 (Equation
mm for wedge
by
appropriate
boundary
and
initial
splitting tests and standard prisms with the dimen- conditi
Themm
relation
between thebending
amount of e
sions 40 x 40 x 160
for three-point
water
and
relative
humidity
is called ‘‘
tests. On the two ends obtained by three-point bendisotherm”
if
measured
with
increasing
ing compressive strength of mortar has been deterhumidity
and
‘‘desorption
isotherm”
in th
mined.
their difference
(Xi et al.
All specimens case.
were Neglecting
allowed to harden
for 28 days
the
following,
‘‘sorption
isotherm”
in a humid chamber at T = 20 ± 2 °C and RH >will be
to both
sorption and
desorption c
95 %. At the agereference
of 28 days
compressive
strength
By
the
way,
if
the
hysteresis
and fracture energy have been determined. Part ofof the
would
taken intooven
account,
the specimens wasisotherm
then dried
in abeventilated
at two
relation,
evaporable
water
vs
relative
105 °C for 14 days. At that time constant weight had humi
be used
according
the sign
of the varia
been achieved. After
cooling
down totoroom
temperarelativity
humidity.
The
shape
ture fracture energy has been determined again. Oth- of the
isothermtoforabsorb
HPC iswater
influenced
er samples were allowed
vapourby
af-many p
especially
those
that
influence
extent
ter drying. They were placed in the laboratory and
chemical
and, byin 20turn,
atmosphere, which
can be reactions
characterized
°C determ
structure
and
pore
size
distribution
and 75 % RH for at least two months. Other pre- (waterratio,
cement
chemical
composition,
SF
dried samples were
placed
in liquid
water or
in seacuring
time
and
method,
temperature,
mix
water for 14 days. Then fracture energy has been deetc.).
In thetest
literature
formulatio
termined by wedge
splitting
and byvarious
three-point
found
to
describe
the
sorption
isotherm
bending.
concrete
1994).specimen.
However, in th
Similar tests have
been(Xi
run etonal.mortar
paper
the
semi-empirical
But on the standard prisms made of mortarexpression
hygral pro
Norling
Mjornell
(1997)
is
adopted b
swelling was measured in addition.
Proceedings of FraMCoS-7, May 23-28, 2010
− D (h, T )∇hAND DISCUSSION
4J =RESULTS
(1)
4.1 The proportionality coefficient D(h,T) is called
moisture permeability and it is a nonlinear function
of the relativestrength
humidityas hobtained
and temperature
T (Bažant
Compressive
after different
cur& Najjar
1972).isThe
moisture
mass 2.
balance
requires
ing
conditions
shown
in Figure
The concrete
that the
timehad
of the
water mass strength
per unit
used
for variation
these testinseries
a compressive
volume
of
concrete
(water
content
w
)
be
equal
to the
of 42.5 N/mm at an age of 28 days. After drying
divergence
of
the
moisture
flux
J
compressive strength had increased to 65.4 N/mm .
This corresponds to an increase of 54 %. Part of this
increase
at
− ∂ = ∇ •can
J be explained by further hydration(2)
∂
elevated
temperature but the major part is due to increased surface energy of the nano-particles in hardwaterpaste
content
can be expressed
as the sum
enedThe
cement
afterw removal
of the adsorbed
wa(capillary
water,
water
of
the
evaporable
water
w
ter layers. At 75 % RH humidity
approximately two
e
vapor, andlayers
adsorbed
water)
the non-evaporable
molecular
of water
areand
re-adsorbed.
As a con(chemically
bound) strength
water decreases
wn (Mills
sequence
compressive
again1966,
to a
Pantazopoulo
& Mills
1995).theItstrength
is reasonable
to
. Finally
decreases
value
of 54 N/mm
assume
that
the
evaporable
water
is
a
function
of
even more if the specimens are water saturated. In
relative
h, the
degree
of hydration,
fact
they humidity,
nearly reach
original
strength. αc, and
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
= age-dependent sorption/desorption isotherm
(Norling
Mjonell 1997). Under this assumption and
)a
byPM substituting Equation 1 into Equation 2 one
(
obtains
ht
g
Strength and fracture energy of untreated
concrete and after different hygral conditioning
2
2
w
explicitly
accounts3 for
evolution
shown
in Figure
are the
average
valuesofofhydration
at least
reaction
and SFmeasurements.
content. This Itsorption
isotherm
three
individual
should be
menreads here that the same tests have also been carried
tioned
out with smaller slabs (200 x 200 x 40 mm). As results are essentially the ⎡same and because ⎤of space
limits
these results shall ⎢not be presented here
⎥ but in
(h α α ) = G (α α ) −
+
⎢
⎥data, as
c s
c s From
aweforthcoming
paper.
the
original
∞
(g α
⎢
c − αstrain
c )h ⎥⎦ softenshown in Figure 3, fracture
and
e
⎣ energy
(4)
ing have been determined⎡ by inverse
analysis
(Bret∞ − α )h ⎤
(g α
tschneider et al.K2007).
c
c − ⎥
⎢
(α α ) e
1
,
,
,
1
1
10
10
c s
,
1
t
2
70.00
60.00
50.00
ne 40.00
trs
ev∂ 30.00
∂
si
+∇•(
−sre
p ∂20.00∂
om
c 10.00
we h
h t
D h ∇h ) =
∂w
e α& + ∂we α& + w&
c
s
n
∂α
∂α
c
s
(3)
where ∂we/∂h is the slope of the sorption/desorption
isotherm (also called moisture capacity). The
governing
equationstrength
(Equation
3) mustconditioned
be completed
Figure
2. Compressive
of differently
conby
appropriate
boundary
and
initial
conditions.
crete cubes.
The relation between the amount of evaporable
water
humidity
called ‘‘adsorption
But and
mostrelative
interesting,
the iscompressive
strength
isotherm” even
if measured
increasing
relativity
decreases
more if thewith
specimens
are saturated
humidity
and ‘‘desorption
in the opposite
with
seawater.
Water has aisotherm”
surface tension
of 72.6
case. Neglecting
et al.tension
1994), of
in
mN/m
(Vargaftik their
et al.difference
1983) and(Xi
surface
the following,
‘‘sorption
isotherm”
used with
seawater
has been
measured
to be will
74.5bemN/m
(see
reference
bothmeans
sorption
desorption
Table
3). to
That
the and
surface
energy conditions.
of the ceBy
the
way,
if
the
hysteresis
of
the
moisture
ment gel is even more reduced after saturation
with
isotherm would be taken into account, two different
seawater.
relation, evaporable water vs relative humidity, must
Table
3. Surface
tensiontoofthe
water,
%
be used
according
signaqueous
of the solutions
variationwithof 3the
and
5 % NaClhumidity.
and seawater.The shape of the sorption
relativity
Liquid for HPC is influenced
Surfaceby
tension
isotherm
many parameters,
mN/m
especially
those
that
influence
extent
and rate of the
Water
chemical
reactions and, 72.6
in turn, determine pore
3 % NaCl and pore size distribution
73.3
structure
(water-to-cement
5
%
NaCl
73.9
ratio, cement chemical composition, SF content,
Seawater
74.5
curing
time and method, temperature,
mix additives,
etc.). In the literature various formulations can be
In Figure
3 thethe
force-CMOD
diagrams
as obfound
to describe
sorption isotherm
of normal
tained
by
wedge
splitting
tests
on
concrete
precondiconcrete (Xi et al. 1994). However, in the present
tioned
same way as described
above
in context
paper inthethesemi-empirical
expression
proposed
by
with
the compression
tests isare adopted
shown. The
curvesit
Norling
Mjornell (1997)
because
0.00
Value
28 days
Dry
RH 75%
Water Sat
Seawater Sat
42.52
65.39
53.98
43.29
39.61
Proceedings of FraMCoS-7, May 23-28, 2010
1
1
1
⎢
⎣
⎥
⎦
where the first term (gel isotherm) represents the
physically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillary
water. This expression is valid only for low content
of SF. The coefficient G1 represents the amount of
water per unit volume held in the gel pores at 100%
relative humidity, and it can be expressed (Norling
Mjornell 1997) as
c α c+ ks α s
G (α c α s ) = k vg
c vg s
(5)
,
1
c Force-CMOD
s
Figure
diagrams as obtained by the wedge
where3.ktest
vg and k vg are material parameters. From the
splitting
(WST)
from
pre-conditioned
in that
the same
maximum
amount
of concrete
water
per
unit volume
can
way
as specimens
prepared
for compression
tests (see Fig.
2).
fill all pores (both capillary pores and gel pores), one
as one as
obtains
canThe
calculate
fractureK1energy
obtained by inverse analysis of the force-CMOD diagrams
shown in Figure
3
⎛
⎞ ⎤
∞
− α fracture
are compiled in Table 4. It can⎡⎢ be seen
⎜ g α that
⎟h ⎥
w −
α increases
G ⎢ − e ⎝ c byc ⎠drying.
⎥
energy of concrete
c s + α s s − considerably
⎥
⎢
The
of fracture energy⎣ by drying is slightly
⎦ (6)
K (α cincrease
α )=
s
⎛
⎞
∞
more than 20 %. But due
to adsorption of water on
⎜ g α − α ⎟h
c ⎠ − decreases again
the internal surface fracture
e ⎝ c energy
after storage in air with RH =c75 %. The
fracture enksvg and Asg1 one
can
parameters
k vg and
ergyThe
is material
further reduced
by water
saturation.
be
calibrated
by
fitting
experimental
data
relevant
to
could have expected saturation with seawater refree (evaporable)
content
concretewithat
duces
fracture energywater
even more
thaninsaturation
variousasages
(Di Luzio
& Cusatis
2009b).is higher see
water
the surface
tension
of seawater
Table 3).
2.2 Temperature
evolution
Table
4. Fracture energy
Gf in J/m2 as determined by wedge
splitting
test at
(WST)
by three-point
(3PB)
on difNote that,
earlyandage,
since the bending
chemical
reactions
ferently
pre-conditioned
concrete
samples.
associated with cement hydration and SF reaction
Moist.
28 days the
dried
75 % field
Water
Seaare exothermic,
temperature
is not uniform
Cond.
RH
Sat.
water
for non-adiabatic
systems even
WST
128
154
145 if the
118environmental
106
temperature
Heat conduction
can be
3PB
110 is constant.
154
131
98
93
described in concrete, at least for temperature not
exceeding
100°C obtained
(Bažant by&theKaplan
1996),
by
In
Figure 4 results
three-point
bendFourier’s
whichThe
reads
ing
test arelaw,
shown.
tendency is the same as already discussed in context of the results of the wedge
q = − λ ∇Ttest shown in Figure 3. From the forcesplitting
(7)
deflection diagrams shown in Figure 4 the fracture
energy
strain
has also
where and
q is
thesoftening
heat flux,
T been
is thedetermined.
absolute
All
results are and
compiled
in Table
The surface energy
temperature,
λ is the
heat 4.conductivity;
in this
increases by 40 % by drying the specimens. But it de10
0
0.188
0.22
1
1
,
1
10
1
1
1
creases below the initial value obtained after 28 days
wet curing. This is pro ably due to damage induced
into the material during the drying process.
28 days wet curing concrete
Totally dry concrete
75% RH re-humidified concrete
Water saturated oncrete
Seawater saturated oncrete
1600
)
N
( 1400
e
c
r 1200
o
F
1000
seawater saturated concrete
treated by silane gel,
)
N
(
e 1000
c
r
o
F
then water saturated
treated by silane gel,
then seawater saturated
800
600
400
− ∂ = ∇•J
∂
w
200
800
t
0
600
The water content w can be expressed a
of the evaporable
CMOD (mm) water we (capillary wa
vapor,
and adsorbed
and the non-e
Figure 5. Force-CMOD
diagram
of neat andwater)
water repellent
bound) water wn (Mil
concrete immersed in(chemically
water and seawater.
Pantazopoulo & Mills 1995). It is reas
Table 5. Fracture energy
in J/m2
on untreated
and is a fu
assume
thatas measured
the evaporable
water
water repellent concrete
samples
immersed
in
water
and
in
relative humidity, h, degree of hydration
seawater.
degree of silica fume
Untreated
Waterreaction,
repellent αs, i.e. we=w
=
age-dependent
sorption/desorption
Water
97.5
115.4
(Norling Mjonell 122.0
1997). Under this assum
Seawater
93.3
by substituting Equation 1 into Equati
obtainsstrength of similar treated
termined compressive
concrete samples.
0.0
400
200
0
0.0
water saturated concrete
The proportionality coefficient D(h,T)
moisture permeability and it is a nonlinea
of the relative humidity h and temperature
& Najjar 1972). The moisture mass balanc
that the variation in time of the water mas
volume of concrete (water content w) be eq
divergence of the moisture flux J
1400
1200
1800
= − D ( h , T ) ∇h
J
1600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Deflection (mm)
Figure 4. Force-deflection diagram as obtained by three-point
bending test (3PB) on differently pre-conditioned concrete.
4.2 Fracture energy of mortar after different hygral
conditioning
Concrete has been dried at an age of 28 days at
105 °C. After having cooled down to room temperature the specimens were immersed in water and in
seawater respectively for 14 days. Some specimens,
however, were surface impregnated with liquid silane before immersion in order to make the surface
near zone water repellent. Then fracture energy has
been determined in this case on small specimens.
The force-CMOD diagrams as obtained by wedge
splitting tests are shown in Figure 5. From these data
fracture energy and strain softening has been determined by inverse analysis. The initial fracture en-2
ergy after 28 days wet curing was about 1002 J/m .
This value increased by drying to 127 J/m . The
fracture energy measured on untreated concrete
samples after immersion in water and seawater and
the corresponding values measured on water repellent samples are compiled in Table 5. It can be seen
that the fracture energy is strongly reduced in the untreated samples by immersion in water or seawater.
Water repellent treatment prevents rapid water absorption by capillary action. But as the pores are not
blocked water vapour can penetrate nevertheless.
The adsorbed water and capillary condensed water reduces gradually fracture energy but if the concrete surface is in contact with water or seawater
temporarily only water repellent treatment provides
a certain protection. Strength and fracture energy
will be less reduced after water repellent surface impregnation.
These results agree well with experimentally de-
0.1
4.3 Fracture
0.2
0.3
0.4
0.5
0.6
0.7
∂w ∂h
e + ∇ • ( D ∇h) = ∂we α ∂we
c
h different
∂α
∂α
t
energy∂hof ∂mortar
after
hygral
−
c
conditioning
&+
s
α&s + w
/∂h is the
of the
sorption/
where ∂weabove
Similar tests as described
for slope
concrete
have
moisture
been carried out isotherm
on cement(also
mortarcalled
specimens.
The capac
governing
equation
(Equationhygral
3) must be
compressive strength
obtained
after different
by
appropriate
boundary
and
initial
conditioning is shown in Figure 6. It can be seen that conditi
relation
between have
the amount
the same changes asThe
observed
on concrete
been of e
waterDrying
and relative
humidity
called ‘‘
measured on mortar.
increases
strengthis by
isotherm” if measured
increasing
54 % but after re-humidification
strength with
decreases
humidity
andtheoretical
‘‘desorption
isotherm” in th
again. As expected
from the
background
case.strength
Neglecting
theirisdifference
(Xi et al.
most of the observed
increase
reduced already after storagetheatfollowing,
75 % RH. ‘‘sorption isotherm” will be
reference to both sorption and desorption c
By the way, if the hysteresis of the
isotherm would be taken into account, two
)a
P
relation, evaporable water vs relative humi
M
h(t
be used according to the sign of the varia
gn
ret
relativity humidity. The shape of the
S
ev
isotherm for HPC is influenced by many p
is
se
especially those that influence extent and
r
m
oC
chemical reactions and, in turn, determ
structure and pore size distribution (waterratio, cement chemical composition, SF
curing time and method, temperature, mix
etc.).
In the
literature
various
formulatio
Figure 6. Compressive
strength
of mortar
at an age
of 28 days
found
to
describe
the
sorption
isotherm
and after different hygral conditioning.
concrete (Xi et al. 1994). However, in th
expression
This is due to paper
the factthethatsemi-empirical
the first adsorbed
lay- pro
adopted b
ers reduce surfaceNorling
energy Mjornell
of the gel(1997)
particlesis most
45.00
40.00
35.00
30.00
25.00
20.00
15.00
10.00
5.00
0.00
Value
28 days
Dry
RH=75%
Water Sat
Seawater Sat
28.09
43.19
33.70
30.50
29.50
Proceedings of FraMCoS-7, May 23-28, 2010
J = − D ( h, T )∇h(see Fig. 1). Again it is observed that
significantly
(1)
strength of specimens saturated with seawater,
which
a higher surface
tension than
tap is
water,
is
Thehas
proportionality
coefficient
D(h,T)
called
further
reduced.
moisture permeability and it is a nonlinear function
energy
of mortar
after different
hygral
of Fracture
the relative
humidity
h and temperature
T (Bažant
conditioning
has
been
measured
the
same
way
as on
& Najjar 1972). The moisture mass balance requires
concrete
samples
by
the
wedge
splitting
test
and
by
that the variation in time of the water mass per unit
three
point
bending.
The
force-CMOD
diagram
is
volume of concrete (water content w) be equal to the
shown
in Figure
and the flux
force-deflection
diagram
divergence
of the7moisture
J
is shown in Figure 8.
− ∂ = ∇•J
∂
1
,
,
,
t
The water content w can be expressed as the sum
of the evaporable water we (capillary water, water
vapor, and adsorbed water) and the non-evaporable
(chemically bound) water wn (Mills 1966,
Pantazopoulo & Mills 1995). It is reasonable to
assume that the evaporable water is a function of
relative humidity, h, degree of hydration, αc, and
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
= age-dependent sorption/desorption isotherm
(Norling Mjonell 1997). Under this assumption and
by substituting Equation 1 into Equation 2 one
obtains
∂w ∂h
e + ∇ • ( D ∇h) = ∂we
h
∂α
∂h ∂t
∂w
α&c + e α&s + w&n
∂α
c
s
(3)
where ∂we/∂h is the slope of the sorption/desorption
isotherm (also called moisture capacity). The
governing equation (Equation 3) must be completed
by appropriate boundary and initial conditions.
The relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
isotherm” if measured with increasing relativity
humidity and ‘‘desorption isotherm” in the opposite
case. Neglecting their difference (Xi et al. 1994), in
Figure
7. Force-CMOD
diagram
as measured
samthe following,
‘‘sorption
isotherm”
willonbemortar
used with
ples,
which
were
differently
conditioned
before
the
test.
reference to both sorption and desorption conditions.
By the way, if the hysteresis of the moisture
From Figures
fracture
energy
strain
isotherm
would be7 and
taken8 into
account,
twoand
different
softening
have been water
determined
by inverse
analysis
relation, evaporable
vs relative
humidity,
must
(see
Brettschneider
et
al.
2007).
The
fracture
be used according to the sign of the variationenergy
of the
as
obtainedhumidity.
on the wetThe
curedshape
specimens
at ansorption
age of
relativity
of the
28
days,
after
drying
at
105
°C
for
14
days,
after
reisotherm for HPC is influenced by many parameters,
humidification
an environment
withand
RHrate
= 75
%,
especially thoseinthat
influence extent
of the
after
water
saturation
and
after
saturation
with
seachemical reactions and, in turn, determine pore
water
is compiled
Table
5. The trend
observed on
structure
and pore insize
distribution
(water-to-cement
concrete
samples
after
hygral
preconditioning
and
ratio, cement chemical composition, SF content,
described
is the same
for mortar.
are
curing timeabove
and method,
temperature,
mixResults
additives,
confirmed.
etc.). In the literature various formulations can be
In allto cases
it could
be shownisotherm
that strength
and
found
describe
the sorption
of normal
fracture
energy
are
highest
in
the
dry
state.
The
surconcrete (Xi et al. 1994). However, in the present
face
of the cement gel
is at maximum
under
paperenergy
the semi-empirical
expression
proposed
by
these
conditions.
Few
adsorbed
water
layers
lowerit
Norling Mjornell (1997) is adopted because
the surface energy significantly. By water saturation
Proceedings of FraMCoS-7, May 23-28, 2010
1
1
10
1
⎡ 10(g α ∞
K1 (α c , α s )⎢e 1 c
⎢
⎣
(2)
w
−
explicitly
the evolution
the
surfaceaccounts
energy isfor
further
lowered butof byhydration
a comreaction and
content.This
Thiscansorption
isotherm
paratively
smallSFamount.
be explained
by
reads
the
relation shown in Figure 1. If concrete or mortar
samples are saturated with seawater, strength and
fracture energy are lowered
even more than ⎤by satu⎡
ration
with
tap
water
as
the
⎢ surface energy of ⎥seawater
(h α α ) = G (α α ) −
+
⎢
⎥
c asscompared
c tos surface
iswehigher
energy
of
water.
∞
(g α
⎢
c − α c )h ⎥⎦ (4)
e
⎣
− α c )h
⎤
− 1⎥
⎥
⎦
where the first term (gel isotherm) represents the
physically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillary
water. This expression is valid only for low content
of SF. The coefficient G1 represents the amount of
water per unit volume held in the gel pores at 100%
relative humidity, and it can be expressed (Norling
Mjornell 1997) as
c α c+ ks α s
G (α c α s ) = k vg
(5)
c
vg
s
Figure 8. Force-deflection diagram as obtained by three-point
bending test
on differently preconditioned mortar prisms.
where kcvg and ksvg are material parameters. From the
,
1
maximum
amount
ofofwater
unit volume
thatdiffercan
Table
5. Fracture
energy
mortarper
as determined
under
ent
conditions.
fillhygral
all pores
(both capillary pores and gel pores), one
Hygral
WetK1 as Dried
ReWater Sea
one obtains
can
calculate
Cond.
cur.
105 °C humid. Sat.
water
28days
75 %
Sat.
⎤
⎡
⎛
α ∞ − α ⎞⎟h53.2
⎜ g57.5
WST
62.9
89.3
74.8⎢
⎥
c
c
⎝
⎠
+
w −
α 95.5
s
α s −74.2
G
⎥
⎢ −e
3PB
60.1
60.2
58.6
c
s
⎥
⎢
⎦ (6)
⎣
K (α c α s ) =
⎞
∞ described
Similar results have ⎛⎜ gbeen
in
a
recent
α − α ⎟h
c ⎠ −Ogishi (Ogishi et
publication (Wittmanne et⎝ al.c 2007).
10
0
0.188
0.22
1
1
1
,
1
10
1
1
al. 1986) was probably the first
to describe
this phec
s
and
k
and
g1 can
The material
nomenon.
Moreparameters
recently kMatsushita
&
Onoue
vg
vg
be calibrated
by fitting
relevant
to
(2006)
have studied
theexperimental
influence of data
surface
energy
freestrength
(evaporable)
water
concrete at
on
under static
and content
dynamic in
loading.
various
ages (Dinano-cracks
Luzio & Cusatis
All
individual
in the2009b).
xero-gel can be
formed with less energy if the surface is covered
with
a water layer,evolution
hence at lower stress level. As
2.2
Temperature
mentioned before, the integral over the energy conNote that,ofatallearly
age, since
the leads
chemical
sumption
individual
events
to thereactions
macroassociated
with energy.
cement hydration
and in
SF the
reaction
scopic
fracture
This depends
first
are exothermic,
the temperature
field
nottheuniform
place
on the moisture
condition but
alsois on
stress
for non-adiabatic
systems
evenNano-cracks
if the environmental
distribution
in a given
sample.
are preftemperaturebeing
is constant.
conduction
be
erentially
formed inHeat
the volume
undercanhigh
described in concrete, at least for temperature not
stress.
exceeding 100°C (Bažant & Kaplan 1996), by
Fourier’s law, which reads
4.4
q = − λ ∇T
(7)
The influence of change of surface energy on
strength
brittleflux,
material
be described
where qof isan ideal
the heat
T can
is the
absolute
by
means
of
the
Griffith
criterion
(see
Equation
In
temperature, and λ is the heat conductivity; (1)).
in this
cement-based materials macroscopic fracture energy
Fracture energy, strength, and hygral length
change
is the sum of a huge number of elementary processes, which can be described by means of Equation
(1) and therefore fracture energy, can be assumed to
be proportional to surface energy. The influence of
change of surface energy on hygral length change is
given by the Bangham equation (see Equation (3))
as predicted by the Munich model (Wittmann 1976,
1977). By combining Equation (1) with Equation (3) a
relation between strength decrease and hygral length
change is obtained (see Equation (4)).
In Figure 9 the square of the bending strength and
the square of the compressive strength [N2/mm4] as
obtained on samples saturated with water, aqueous
solutions with 3 % and 5 % of NaCl and seawater
are shown. Depending on the surface energy of these
liquids the surface energy of mortar is reduced. The
reduction of surface energy obviously leads to a reduction of both bending and compressive strength.
It should be mentioned at this point that the interpretation presented in this contribution is a simplification of the real situation, primarily because the influence of disjoining pressure has not been taken
into consideration separately Wittmann 1976, 1977).
Figure 9. Square of compressive and bending strength as function of the swelling strain after saturation with water, aqueous
solutions with 3 and 5 % NaCl and seawater.
The influence of surface energy on strength and
swelling is dominant in the humidity range between
0 % and approximately 70 %. In the case of saturation, however, the influence of dissolved ions on
disjoining pressure and hence on strength and
swelling cannot be neglected (Wittmann 1976, Setzer 2007). To quantify this effect further tests are
necessary.
5 CONCLUSIONS
It has been shown that strength and fracture energy
depend on surface energy of the hydration products
of cement in cement based materials such as con-
= − D (h, T )energy
∇h
crete and mortar.J Surface
can be changed
significantly by changing the hygral conditions. Surface energy and strength
as well as fracture
energy D(h,T)
The proportionality
coefficient
are at maximum moisture
in the completely
dry
state.
One
permeability and it is aornonlinea
two adsorbed wateroflayers
reducehumidity
surface energy
the
the relative
h and of
temperature
hydration products& significantly.
This
leads
to
a
re- balanc
Najjar 1972). The moisture mass
markable reductionthat
of strength
and
fracture
energy.
the variation in time of the water mas
Fracture energyvolume
can beofconsidered
to be content
the sumw) be eq
concrete (water
of a huge numberdivergence
of small quantities
of surface
of the moisture
flux Jenergies consumed by formation of individual nanocracks in the xero-gel.
− ∂ = ∇with
• J hydration products reSeawater in contact
∂
duces surface energy even more than pure water as
the surface tension ofThe
seawater
is bigger.
Thisbeeffect
water content
w can
expressed a
has to be taken into
consideration
if
crack
of the evaporable water formation
we (capillary wa
in reinforced concrete
situated
in marine
vapor,structures
and adsorbed
water)
and the non-e
environment is to(chemically
be predicted.bound)
Durability
of conwater
wn (Mil
crete structures depends
strongly
prevention
Pantazopoulo
& on
Mills
1995). Itof is reas
crack formation. assume that the evaporable water is a fu
The reduction relative
of strength
and fracture
energy
humidity,
h, degree
of of
hydration
concrete in temporary
withfume
water
or seawater
degreecontact
of silica
reaction,
αs, i.e. we=w
as for instance in=the age-dependent
splash zone can sorption/desorption
be limited by
water repellent surface
treatment.
(Norling Mjonell 1997). Under this assum
The observed by
behavior
has beenEquation
predicted1 byinto
the Equati
substituting
Munich Model (Wittmann
obtains 1976, 1977)). On this basis the reduction of fracture energy can also be
linked with hygral length
such as∂swelling.
A
∂w
we
we ∂hchange
strong correlation− ∂between
strength
decrease
and
α&c + e α&s + w
+ ∇ • ( D ∇h ) =
h
∂
α
∂α
∂
∂
h
t
swelling has been confirmed.
c
Reduction of strength and fracture energy of ce-s
ment based materials
of eparticular
/∂h
isreinforced
the importance
slope of
thefor
sorption/
whereis ∂w
durability and service
life
of
concrete
isotherm
(also
called
moisture
capac
structures in marine
environment.
governing
equation (Equation 3) must be
by appropriate boundary and initial conditi
The relation between the amount of e
ACKNOWLEDGEMENT
water and relative humidity is called ‘‘
if measured
with increasing
Authors gratefullyisotherm”
acknowledge
financial support
of in th
humidity
and
‘‘desorption
isotherm”
this project by National
Natural
Science
Foundation
Neglecting their difference (Xi et al.
of China, contractcase.
No.following,
50378045.
the
‘‘sorption isotherm” will be
reference to both sorption and desorption c
the way, if the hysteresis of the
REFERENCES By
isotherm would be taken into account, two
relation,
relative humi
Alvaredo, A. M. 1994.
Drying evaporable
shrinkage andwater
crack vs
formation.
be
used
according
to
the
sign
of the
In F. H. Wittmann (ed.) Building Materials Reports No.
5, varia
relativity
humidity.
The
shape
of the
Aedificatio Publishers Freiburg.
Bangham, D. H. & Fakhoury
J.
1931.
The
swelling
of
charcoal
isotherm for HPC is influenced by many p
I: Preliminary experiment
waterthat
vapour,
cabon dioxespeciallywiththose
influence
extent and
ide, monia, and sulphur
dioxide.
Proc.
Royal
Soc.
Aturn,
130: determ
chemical
reactions
and,
in
81-89.
structure
pore
distribution
Bangham D. H. & Razouk
R. I.,and
1937.
(a) size
Adsorption
and the (waterratio,
cement
chemical
composition,
SF
wettability of solid
surfaces.
Transactions
Faraday
Soc. 33:
and method,
temperature,
1459; and (b) Thecuring
wettingtime
of charcoal
and the nature
of the mix
adsorbed phase formed
vapours.
Transacetc.). Infromthesaturated
literature
various
formulatio
tions Faraday Soc.found
33: 1463.
to
describe
the
sorption
isotherm
Brettschneider N., Villmann
B.
&
Slowik
V.
2007.
Inverse
concrete
et al. 1994).
However,
analysis of bending
tests for(Xi
determining
the properties
of in th
paper
the
semi-empirical
expression
pro
strain hardening cement-based materials. In Carpinteri et al.
Norling ofMjornell
(1997)
adopted b
(eds) Fracture Mechanics
Concrete and
Concreteis Strucw
t
Proceedings of FraMCoS-7, May 23-28, 2010
J
=tures
− D (h-, THigh-performance
) ∇h
concrete. Taylor & Francis
(1)
Group, London. P. 1419-1427.
Carpinteri, A., Gambarova, P., Ferro, G. & Plizzari, G. (Eds.)
The proportionality
is called
2007.
Fracture Mechanicscoefficient
of Concrete D(h,T)
and Concrte
Strucmoisture
permeability
and it isanda Francis,
nonlinear
function
tures, Proc.
FRAMCOS 6.Taylor
London.
Hillerborg,
A. 1985.
The theoretical
basis of a method
to deof the relative
humidity
h and temperature
T (Bažant
termine
fracture
energy
G
of
concrete.
Materials
and
Struc& Najjar
1972).
The moisture mass balance requires
tures,
18:
291-296.in time of the water mass per unit
that
the
variation
Hillerborg, A., Modeer, M. & Petersson, P. E. 1976. Analysis
volume
of formation
concreteand
(water
be equal
the
of crack
crackcontent
growth inw)concrete
by to
means
divergence
of
the
moisture
flux
J
of fracture mechanics and finite elements. Cem. Concr.
Res. 6: 773-782.
Matsushita,
H. & Onoue, K. 2006. Influence of surface energy
w
− ∂on
=compressive
∇•J
strength of concrete under static and (2)
dyt
∂namic
loading. J. Advanced Concrete Technology 4: 409421.
TheS.,water
w can be
expressed
asofthesurface
sum
Ogishi,
Ono, content
H. & Tanahashi,
I. 1986.
Influence
of several water
kinds ofwe impregnated
the
(capillary liquids
water, onwater
of tension
the evaporable
strength
and
Young’s
modulus
of
cement
mortar.
In
F.
vapor,
and (ed.)
adsorbed
water)
and and
the Fracture
non-evaporable
Wittmann
Fracture
Toughness
Energy
of
(Mills P.1966,
(chemically
bound)
wn Amsterdam.
Concrete. Elsevier
Sciencewater
Publishers,
Pantazopoulo
&
Mills
1995).
It
is
reasonable215to
218.
assume
that
the
evaporable
water
is
a
function
of
Setzer M. J., 1973. Oberflächenenergie und mechanische
Eigenschaften
des hZementsteins,
Thesis, αMunich
relative
humidity,
, degree of PhD
hydration,
c, and
University
of
Technology
(TUM).
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
Setzer
M. J., 2007. The solid-liquid
gel-system of isotherm
hardened
= cement
age-dependent
sorption/desorption
paste. In M. J. Setzer (ed.) Transport in Concrete:
(Norling
1997). Under
and
Nano- toMjonell
Macrostructure,
Proc. 5 this
Int. assumption
Essen Workshop,
by Aedificatio
substituting
Equation
1 Germany.
into Equation
Publishers
Freiburg,
P. 3-23. 2 one
obtains N. B., Volkov, B. N. & Voljak, L. D. 1983. InternaVargaftik,
tional Tables oft he Surface Tension of Water. J. Phys.
∂w
∂w
∂Chem.
w Ref. Data 12(3): 317-320.
f
th
−
e ∂h + ∇ • ( D ∇h) = e α& + e α& + w&
c
s
n
h
∂α
∂α
∂h ∂t
c
s
(3)
where ∂we/∂h is the slope of the sorption/desorption
isotherm (also called moisture capacity). The
governing equation (Equation 3) must be completed
by appropriate boundary and initial conditions.
The relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
isotherm” if measured with increasing relativity
humidity and ‘‘desorption isotherm” in the opposite
case. Neglecting their difference (Xi et al. 1994), in
the following, ‘‘sorption isotherm” will be used with
reference to both sorption and desorption conditions.
By the way, if the hysteresis of the moisture
isotherm would be taken into account, two different
relation, evaporable water vs relative humidity, must
be used according to the sign of the variation of the
relativity humidity. The shape of the sorption
isotherm for HPC is influenced by many parameters,
especially those that influence extent and rate of the
chemical reactions and, in turn, determine pore
structure and pore size distribution (water-to-cement
ratio, cement chemical composition, SF content,
curing time and method, temperature, mix additives,
etc.). In the literature various formulations can be
found to describe the sorption isotherm of normal
concrete (Xi et al. 1994). However, in the present
paper the semi-empirical expression proposed by
Norling Mjornell (1997) is adopted because it
Proceedings of FraMCoS-7, May 23-28, 2010
Wittmann
H., 1976. The
of hardenedofcement
paste
explicitlyF. accounts
forstructure
the evolution
hydration
–
A
basis
for
better
understanding
of
the
material
properreaction and SF content. This sorption isotherm
ties. In Proc. Conf. Hydraulic cement pastes: their structure
reads
and properties, Cement and Concrete Association, Wexham
Springs. P. 96-117.
Wittmann F. H., 1977. Grundlagen
eines Modells
zur
⎡
⎤
Beschreibung charakteristischer
Eigenschaften des
⎢
⎥ Betons,
(h α α )Ausschuss
weDeutscher
−
+ 43Report
No.⎥ 290:
c s = G (α c αfürs )⎢Stahlbeton,
∞
(g α
− α )h ⎥
101.
⎢
e T.,c 2006.c Influence
⎦
(4)of
Wittmann, F. H., Zhang, P.⎣& Zhao
combined environmental loads
on durability
∞ − α )h of reinforced
⎡
⎤
(g α
concrete structures.
Int.
J.
Restoration
of
Buildings
and
c
c
⎢
K (α c α s ) e
− ⎥
Monuments 12 (4): 349-362.
⎢
⎥
⎣
Wittmann, F. H., Sun Z. & Zhao
T., 2007. Strength⎦and fracture energy of concrete in seawater. In Carpinteri et al.
(eds) the
Fracture
andrepresents
Concrete Strucwhere
firstMechanics
term (gelof Concrete
isotherm)
the
tures – New Trends in Fracture Mechanics of Concrete.
physically
bound
(adsorbed)
water
and
the
second
Taylor & Francis Group London. P. 213-217.
term (capillary
isotherm)
the capillary
Zaitsev
Y. V. & Wittmann
F. H.,represents
1981. Simulation
of crack
water.
This expression
onlyMaterials
for lowandcontent
propagation
and failure is
of valid
concrete.
Structures
14:
357-365.
of SF. The coefficient G1 represents the amount of
Wittmann,
F.unit
H., volume
Sun Z. &held
ZhaoinT.,the
2007.
Strength
fracwater
per
gel
pores
atand100%
ture
energy
of
concrete
in
seawater.
In
Carpinteri
et
al.
relative
humidity,
and itofcan
be expressed
(Norling
(eds) Fracture
Mechanics
Concrete
and Concrete
StrucMjornell
as in Fracture Mechanics of Concrete.
tures – 1997)
New Trends
Taylor & Francis Group London. P. 213-217.
Zaitsev Y. V. &c Wittmanns F. H., 1981. Simulation of crack
(α α ) = k α c + k α s
G propagation
(5)
and cfailurevgof sconcrete. Materials and Strucc s vg
tures 14: 357-365.
Polhill, R.M.
1982. Crotalaria in Africa and Madagascar. Rotand ksvg are material parameters. From the
where
kcvgBalkema.
terdam:
1
,
,
,
1
1
10
10
,
1
1
1
1
,
1
maximum amount of water per unit volume that can
fill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtains
w
0
K (α c α s ) =
,
1
⎡
⎢
−
⎢1 −
1
⎢
⎣
α s + 0.22α s G
− 0.188
c
s
⎛
10⎜
⎝
e
⎛
10⎜
e
g αc − αc h
−
∞
1
⎞
⎟
⎠
⎝
g α c∞ − α c ⎞⎟h ⎤⎥
1
⎠
⎥
⎥
⎦
(6)
1
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
free (evaporable) water content in concrete at
various ages (Di Luzio & Cusatis 2009b).
2.2 Temperature evolution
Note that, at early age, since the chemical reactions
associated with cement hydration and SF reaction
are exothermic, the temperature field is not uniform
for non-adiabatic systems even if the environmental
temperature is constant. Heat conduction can be
described in concrete, at least for temperature not
exceeding 100°C (Bažant & Kaplan 1996), by
Fourier’s law, which reads
q
= − λ ∇T
(7)
where q is the heat flux, T is the absolute
temperature, and λ is the heat conductivity; in this